So, I am knitting a hat on regular straight needles. It is obviously supposed to turn out that I knit it sideways and sew together the cast on / cast off edges to make the hat round.

The patt. asks me to cast on 56-58-60 sts, then it says:

"work in garter st while AT THE SAME TIME working as follows: *2 rows over all sts on needle, 2 rows over the first 52 sts on needle, 2 rows over the first 44 sts on the needle* repeat from *-*"

I don't know what it means by "knit over," also, I am doing the largest size, but it does not tell me how many sts to knit over for that size, do I just do thesame number as it says in the patt no matter what size I am doing?

You're doing short rows over some of the sts which will shape the hat. So knit the first 52sts for all sizes, turn and knit back on those same 52sts. Then knit 44sts, turn and knit back over the same 44sts and repeat the sequence of rows. The instructions are the same for all sizes for this part of the hat.

It's doing the shaping by short rows where you don't knit all the stitches across every time.

So do 2 rows on all the stitches, then turn and knit 52 sts, turn and knit back over (same as 'on') those stitches. Next row, k 44, turn and knit back over the same sts. On the next row you work that sequence again - knit all sts for 2 rows, knit 52 sts for 2 rows, k44 for 2 rows. The size of it will be determined by how many rows you do, not by the number of stitches.

As you're new to short rows, a video might be good. Does the pattern say to wrap and turn, or to slip a stitch? Can you link to the pattern? As there are many short row methods, sending you to watch a video for a method other than your pattern is using would be confusing.

__________________~ GG
Happiness is catching that dropped stitch while it's still fixable..